Robin-to-Neumann and Dirichlet convergence conjecture for the porous medium equation

Let $T>0$, let $m$ be the porous-medium exponent, and let $g:{\mathbb T}\to[0,1]$ be measurable. For each $\kappa>0$, let $\rho^\kappa:[0,T]\times{\mathbb T}\backslash\{0\}\to[0,1]$ be the unique weak solution of the porous medium equation with Robin boundary conditions and initial condition $g$. Let $\rho^0$ be the unique weak solution with Neumann boundary conditions, and let $\rho^\infty$ be the unique weak solution with periodic boundary conditions, both with initial condition $g$. Robin-to-Neumann and Dirichlet convergence conjecture. In $L^2([0,T]\times{\mathbb T}\backslash\{0\})$, one has

$$\lim_{\kappa\to0}\rho^\kappa=\rho^0$$

and

$$\lim_{\kappa\to\infty}\rho^\kappa=\rho^\infty.$$

The claim describes the limiting behavior of the Robin boundary condition as its parameter tends to zero or infinity, but the source gives no resolution or proof of this convergence statement.