Determine whether the multifractal lower bound is always sharp

Determine whether, for general Gatzouras--Lalley measures with separation of principal projections, the lower bound f_{\bm p}(\alpha) \geq \max\{v(\bm w):u_{\bm p}(\bm w)=\alpha\} is always an equality, or whether it can be strict for some values of \alpha.

Background

For a Gatzouras--Lalley measure \mu_{\bm p} with separation of principal projections, the paper establishes the general lower bound f_{\bm p}(\alpha) \geq F_{\bm p}(\alpha), where F_{\bm p}(\alpha)=\max{v(\bm w):u_{\bm p}(\bm w)=\alpha}. Here, f_{\bm p} is the Hausdorff dimension of the set of points having local dimension \alpha, while v(\bm w) is the Hausdorff dimension associated with the Bernoulli measure determined by \bm w and u_{\bm p}(\bm w) is its typical local dimension with respect to \mu_{\bm p}.

The paper proves that this lower bound is exact when the Gatzouras--Lalley system has two columns, using the one-dimensional structure of the column-probability simplex. It explicitly leaves unresolved whether equality holds for general systems with more than two columns, or whether strict inequality can occur. Resolving this issue would determine whether the variational formula for the multifractal spectrum extends beyond the two-column case.