Conjecture on biholomorphic Loewner chains in reflexive Banach spaces

Let $X$ be a reflexive complex Banach space with unit ball $\mathbb{B}$, let $A\in L(X)$, and write $k_+(A)$ and $m(A)$ for the quantities used in the source. An $A$-normalized univalent subordination chain $f(z,t)$ is obtained as the limit

$$\lim_{t\to\infty}e^{tA}v(z,s,t)=f(z,s).$$

Here $v$ is the solution of the associated initial value problem, and a standard solution satisfies the Loewner PDE outside a null set of times.

Biholomorphic-chain conjecture. If $k_+(A)<2m(A)$ and $f(z,t)$ is the $A$-normalized univalent subordination chain given by the displayed limit, then $f(\cdot,t)$ is biholomorphic on $\mathbb{B}$ for every $t\geq 0$, and $f(z,t)$ is a standard solution of

$$\frac{\partial f}{\partial t}(z,t)=Df(z,t)h(z,t).$$

The conjecture concerns the extension of finite-dimensional Loewner PDE results to infinite-dimensional reflexive Banach spaces. The source notes that it is true when $X=\mathbb{C}^n$, while the general infinite-dimensional case remains open.