Objective collapse of wavefunction energy equivalence

HUP alone does not imply any zero point energy.

Non-zero value of zero point energy is a consequence of defining the possible values of energy as eigenvalues of the Hamiltonian

$$ H = \frac{1}{2m}p^2+ \frac{1}{2}m\omega^2x^2 $$ instead of the Hamiltonian

$$ H_{no} = \frac{1}{2m}p^2+ \frac{1}{2}m\omega^2 x^2 - \frac{1}{2}\hbar \omega. $$ It's just a formal difference in description, a kind of "gauge freedom", with zero implication on our predictions of the physical behaviour.

When wave function is localized to a very sharp peak, its energy is undefined, but average value and its uncertainty can be calculated. This average and uncertainty can be much greater than the usual zero point energy $\frac{1}{2}\hbar \omega$.

Since many quantum states do not have definite energy during their evolution in time, no energy conservation can be formulated to hold continuously in any single case; the particle may start with definite energy $E_1$, due to interaction with external field, go through states with undefined energy, and end up found with energy $E_2$ larger than $E_1$. To have the idea of energy conservation at least in some form, we may talk about average ("expectation") value of energy $\langle H \rangle$, which obeys $\partial_t \langle H \rangle = 0$ in case of a time-independent Hamiltonian, and no collapse taking place.

In any individual case of finding a definite value of energy, the surplus energy may be hand-waved to come from the environment - the measurement apparatus, the background radiation, etc.

I dont see the issue with objective collapse theories,in fact it solves many of the mysteries of QM.

They introduce some new assumptions (how often, how fast the collapse happens), but so far didn't bring any great revelation about the world. Details of the collapse they propose are not experimentally verified. Their predictions are in principle experimentally testable, so maybe someone in the future will show that details of collapse happen in time like those theories propose, and we will measure those parameters (how often, how fast); but so far, we don't know much more about the collapse than what the orthodox quantum theory says.

In classical physics the evolution of a measurable quantity $x$ is described by a function $x(t)$ such that if you measure $x$ at time $t$ you get the result $x(t)$.

In quantum physics the evolution of a measurable quantity is described by a matrix. The eigenvalues are the possible results of measuring that quantity and quantum theory predicts the probability of each outcome.

In many experiments the probabilities of the outcomes are affected by what happens in all of the possible states: this is called interference. For an example, see section 2 of this paper:

https://arxiv.org/abs/math/9911150

But in everyday life when you walk through a doorway you don't see effects from all of the possible ways you might have walked through the door. Some physicists proposed that all of the possible outcomes except one go away somehow and called the alleged process that does this collapse. It was and is very rare for people talking about collapse to describe how it actually works, but some physicists are working on theories that explicitly include collapse:

https://arxiv.org/abs/2310.14969

Quantum theory itself predicts that when you copy information out of a quantum system interference is suppressed. This is called decoherence:

https://arxiv.org/abs/1911.06282

Any object you see in everyday life has information copied out of it on scales of space and time smaller than those over which they change significantly so on those scales interference is negligible. Note that decoherence is part of quantum theory without collapse so collapse is unnecessary for decoherence.

Also, decoherence is a process that takes place over time and it can be used to describe unsharp, continuous and repeated measurements that are common in real labs. Collapse theories don't currently do this

https://arxiv.org/abs/1911.06282

It is true that collapse theories has a low probability to kick a particle so that it is far from the peak of the probability and this may have measurable effects:

https://philsci-archive.pitt.edu/11350/

https://arxiv.org/abs/1407.4746

It should also be noted that almost all experimentally tested predictions of quantum theory involve relativistic effects and collapse theories don't currently reproduce those predictions:

https://arxiv.org/abs/2205.00568

It is not correct to say the main problem with collapse theories is about particle kicks. Such theories have a wide range of serious unsolved problems, not the least of which is the lack of any explanation of what problem they solve that isn't already solved by decoherence or other mechanisms in quantum theory without collapse.

You claim that collapse many of the mysteries of quantum theory but provide no examples of those mysteries or how they are solved. There is another theory you can look at to understand quantum theory and to understand how the world works: it is called quantum theory.