Exponential puncture-growth conjecture for Brunnian pseudo-Anosov dilatations

Let $S_{g,p}$ be a genus-$g$ surface with $p$ punctures, let $\brun(S_{g,p})$ denote the set of Brunnian pseudo-Anosov mapping classes on $S_{g,p}$, and let $L(\brun(S_{g,p}))$ denote their dilatation lower bound. Exponential puncture-growth conjecture. There exist constants $A,B>0$ so that

$$L(\brun(S_{g,p})) \geq Ap+B$$

for all $p\geq 1$ and any $g$. This predicts that the dilatations of Brunnian pseudo-Anosov mapping classes grow at least exponentially with the number of punctures, a stronger statement than the results established in the paper; its general validity is left open.