Local continuity of weak solutions to the Stefan problem involving the singular $p$-Laplacian

We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: $$\partial_t \beta(u)-\Delta_p u\ni 0\quad\text{ for }\tfrac{2N}{N+1}<p<2,$$ where $\beta$ is a maximal monotone graph with a jump at zero and $\Delta_p$ is the $p$-Laplacian. Moreover, a logarithmic type modulus of continuity is quantified, which has been conjectured to be optimal.