Regularity conjecture for generators of T-continuous semigroups

Let $X$ be a Banach space, let $d4a5=\left\{F_t\right\}_{t\ge0}\subset C^k(\mathcal{D})$ be a semigroup on a domain $\mathcal{D}\subset X$, and let $T$-continuity mean continuity in the topology specified by the paper. Assume that for each $j=1,\ldots,k$, the family $\left\{{F_t}^{(j)}\right\}_{t\ge0}$ is $T$-continuous. The generator regularity conjecture. The strong limit

$$f(x)=\lim_{t\to0^+} \frac1t\left(F_t(x)-x\right)$$

exists uniformly on subsets strictly inside $\mathcal{D}$. Moreover, $f$ is bounded on every set that lies strictly inside $\mathcal{D}$ and belongs to $C^{k-1}(\mathcal{D},X)$. This conjecture extends the preceding differentiability result from $C^1$ semigroups to higher regularity: simultaneous $T$-continuity of the derivatives is expected to ensure existence and $C^{k-1}$ regularity of the infinitesimal generator. The supplied context gives no resolution, so the conjecture is recorded as open.