The intermediate-dimension interpolation conjecture for connected components of Mandelbrot percolation

Let $F^c$ denote the connected-component part of a Mandelbrot percolation limit set. For a function $[?]$ let $[?]$ denote the corresponding intermediate dimension, and define

$$\psi(\delta)=\frac{\log\lvert\log \delta\rvert}{\lvert\log \delta\rvert}.$$

Intermediate-dimension interpolation conjecture. There exists a monotone function $\alpha(s)\colon[0,1]\to[0,\infty)$ such that

$$\left\{\dim^{\Phi} F^c\colon \Phi(\delta)=\delta^{1+\alpha(s)\psi(\delta)}\right\}=\left[\dim_{\mathrm{H}}F^c,\overline{\dim}_{\mathrm{B}}F^c\right].$$

This conjecture predicts that a suitable family of intermediate dimensions recovers the full interval between the Hausdorff and upper box dimensions of the connected-component set, extending the spectrum-interpolation perspective discussed in the survey.