Variance bound for downcrossings in feasible martingale processes

Let $D_{ab}$ denote the number of downcrossings of the interval $[a,b]$ in a $\boldsymbol{0}$-feasible process, where $0<a<b<1$. The process is $\boldsymbol{0}$-feasible in the sense used in the paper, with all component martingales initially at zero.

Downcrossing variance conjecture. For any $\boldsymbol{0}$-feasible process,

$$\operatorname{var}(D_{ab})\leq \left(\frac{1-b}{b-a}\right)^2+\frac{1-b}{b-a}.$$

The bound is suggested by the preceding construction, in which $D_{ab}+1$ has a geometric distribution with parameter $(b-a)/(1-a)$; its variance is the displayed expression. The source presents the inequality as a natural guess, and no resolution is given here.