Let $S$ be the Sierpinski carpet contained in the square $[0,1]^2$. For Lebesgue almost every $a\in [0,1]^2$ and every $\theta\in\mathbb Q$ the intersection of the line $L_{a,\theta}$ with equation $y-x\tan\theta = a$ with $S$ has Hausdorff dimension strictly less than $\dim_H(S)-1 = \frac{\log 8}{\log 3}-1$ [Citation: Manning-Simon, Dimension of slices through the Sierpinski carpet. Trans. Amer. Math. Soc. MR2984058].
What is known about the minimal Hausdorff dimension of the intersection of a line containing a given point with the carpet? Specifically, what bounds are known for $$ d:=\sup_a\{\inf_{\theta}\{\dim_H(S\cap L_{a,\theta})\}\} $$ where, if helpful, the supremum can be taken over a co-null subset of $[0,1]^2$, and there is no need to restrict $\theta$ to being rational.
It seems intuitive that $d$ cannot be less than the dimension of the middle thirds Cantor set $\frac{\log 2}{\log 3}$, and it is less than $\frac{\log 8}{\log 3}-1$ by Remark 1.4 of [Liu-Xi-Zhao, Dimension of intersections of the Sierpinski carpet with lines of rational slope. Proc. Edin. Math. Soc.]
