Bridge-index conjecture for discrete Morse vectors of knots

Let $K$ be a knot with bridge index $b$. Say that a $d$-complex has discrete Morse vector $(c_0,\ldots,c_d)$ if it admits a discrete Morse function with exactly $c_i$ critical faces of dimension $i$. Bridge-index conjecture. The knot $K$ appears in both of the following ways:

1. As a knotted spanning edge in some $3$-ball with discrete Morse vector $(1,b-2,b-2,0)$, if $b\geq 2$.
2. As a $3$-edge subcomplex of some $3$-sphere with discrete Morse vector $(1,b-3,b-3,1)$, if $b\geq 3$.

The claim is presented as a conjectural consequence suggested by the knot-theoretical approach, relating bridge index to the minimum number of critical faces in a discrete Morse function. Its resolution is not specified in the supplied text.