Integral freeness conjecture for cuspidal algebras of imaginary root multiples

Let $\delta$ be the minimal positive imaginary root of the affine ADE root system, let $n\in\mathbb{Z}_{>0}$, and let $k$ be either $F$ or $K$. Write $C_{n\delta,\mathcal O}$ for the corresponding cuspidal algebra over $\mathcal O$, and $C_{n\delta,k}$ for its base change to $k$. Integral freeness conjecture. The algebra $C_{n\delta,\mathcal O}$ should be free over $\mathcal O$ and satisfy

$$C_{n\delta,k}\cong C_{n\delta,\mathcal O}\otimes_{\mathcal O} k.$$

This extends the preceding integral-freeness and base-change result for $C_{n\alpha,\mathcal O}$ with $\alpha\in\Phi_+$ and for $C_{\delta,\mathcal O}$. The statement is presented as the part that remains open in the general case.