Hausdorff dimension of t-adic Littlewood counterexamples

Let $t$ denote the polynomial used in the $t$-adic Littlewood conjecture, and let $f$ be a growth function. Consider the set of counterexamples to $t$-LC with growth function $f$. The Hausdorff-dimension conjecture. This set has Hausdorff dimension zero when $f$ is of the form $\log^{1+\lambda}$ for any $\lambda<1$. The conjecture seeks to improve the known metric result at the critical $\log^2$ growth scale; the status of the proposed improvement is open.