Top end (fully charged balancing) BMS circuits usually work by bypassing a portion of the charging current from the cell or cells that reach terminating voltage first. This only works when cells are well matched and start off with similar charge states. If the reduction in current is not enough to fully stop the "leading": cell charging the cell will continue charging at a reduced rate.
It will quite often be found that the balancing discharge FETs are able to handle a higher balance current than the installed discharge resistors allow. In such cases using smaller resistors to increase balance current to higher but still safe levels scan help. Resistors must be adequately power rated and all components must be heat-sunk adequately.
If the balance circuitry is faulty, as may be the case here, no resistor change helps, of course. Again: starting with well balanced and matched cells is necessary.
While, as others correctly say, working without circuit diagrams and specifications is "a bad idea", engineering often consists of solving under resourced challenges, and this is one such.
The device control IC is shown on the diagram in your linked article as shown here. Balance FETs are at top right. Balance resistors are not shown but must exist in some form. They seem to need to be inside the IC OR a single resistor may be used in the BATM line.
The article states that the BMS does not work at all - ie not just low-balancing but no-balancing.
I can only make an assessment based on experience moderated probability.
This will probably be good enough, but may not be.
Caveat Emptor :-) .
IF the new BMS works and IF the old BMS does not work at all then adding the new one in parallel should be acceptable.
If desired you could remove the existing balance resistors and/or the balance FETS, but, if they do not work at all in your case then leaving them intact is slightly safer.
