Conjecture on the number of ergodic equilibrium states for singular value potentials

Let $(A_1,\ldots,A_N)\in GL_d(\mathbb{R})$ and let $s\in(0,d)\smallsetminus\mathbb{Z}$. For a tuple of matrices, let $\varphi^s$ denote Falconer's singular value function, and call an invariant measure an ergodic equilibrium state for $\varphi^s$ if it is ergodic and attains the pressure associated with this potential. The ergodic equilibrium-state conjecture. The maximum possible number of ergodic equilibrium states of $\varphi^s$ is precisely

$$(d-\lfloor s\rfloor){d \choose \lfloor s\rfloor}=\lceil s\rceil {d \choose \lceil s\rceil}.$$

The conjecture seeks the sharp upper bound for the number of ergodic equilibrium states of Falconer's singular value function; the preceding discussion notes that the standard bound is not optimal in dimension three, while the precise maximum in the stated generality remains open.