Crab bucket non-destabilization conjecture for Legendrian knots

Let $\beta_n$ be the crab bucket Legendrian knot, and for odd $n\geq 5$ choose an orientation such that

$$\operatorname{rot}(\beta_n)=1.$$

Crab bucket non-destabilization conjecture. For all odd $n\geq 5$, $\beta_n$ cannot be negatively destabilized without increasing its mosaic number.

This conjecture was motivated by the crab bucket construction as a proposed infinite family in which stabilization decreases the mosaic number. The stated obstruction to negative destabilization is presented as an apparent phenomenon of the construction, and its general validity remains open.