Charcoal, 3232 18 bytes
⊞υ²ＦＳ⊞⊞Ｏυ№_/ι№⌊⭆θ∨¬κ⁼№_/‾ι⊞υ²⌊⭆⪪υ²›⌈ι⌊ιι№\_§θ⊖κ Try it online!Try it online! Link is to verbose version of code. Would be 30 bytes if I used ~ but ‾ looks nicer. Explanation:
⊞υ² θ Input string Push 2 to the predefined list.
ＦＳ ⭆ Map over characters and convert to string Loop over the input string.
⊞⊞Ｏυ№_ κ Current index ¬ Logical Not (i.e. is zero) ∨ Logical Or ι Current character № Count (i.e. contained in) _/ι№ Literal _/‾ι (i.e. begins at bottom) For each character, push two bits to the list, one for if the character begins at the bottom and one for if it ends at the top.
⊞υ² ⁼ Equals Push another 2 to the list.
⌊⭆⪪υ²›⌈ι⌊ι θ Input string § Indexed by κ Current index ⊖ Decremented (i.e. previous character) № Count (i.e. contained in) \_ Literal \_ (i.e. ended at bottom) ⌊ Minimum (i.e. if all true) Implicitly print Compare pairs of bits for inequality. Because of the extra two 2s, this now matches pairs of bits from adjacent characters. The results are then cast to string and the minimum taken, which means that the output is 0 if any of the pairs are the same (meaning that the string is disconnected) and 1 if they are all different (meaning that the string is connected).