Universal abelian-cover torsion growth conjecture for pseudo-Anosov mapping tori

Let $\psi$ be a pseudo-Anosov mapping class and let $T_\psi$ be its mapping torus. A universal tower of finite abelian covers of a finite CW complex $X$ is a tower of finite abelian covers that eventually factors through every finite abelian cover of $X$. Exponential torsion homology growth means that the torsion in homology grows exponentially along the tower.

Torsion conjecture (T1.5). There exists a finite cover $X$ of $T_\psi$ and a universal tower of finite abelian covers $\{X_i\}$ of $X$ which have exponential torsion homology growth.

This statement is positioned between the two torsion conjectures above and is stated to be logically equivalent to the stronger surface conjecture asserting $\rho(\widetilde{\psi}_*)>1$.