Higher-order differentiability conjecture for spectral functions

Let $X$in the space of symmetric matrices have an ordered spectral decomposition

$$X=V\bigl(\operatorname{Diag}\lambda(X)\bigr)V^T,$$

and let $f$ be a function on the eigenvalue vectors, with $P^k$ denoting the set of partitions of $\{1,\ldots,k\}$ and $T^{k,n}$ the space of $k$-tensors on $\mathbb{R}^n$. For symmetric matrices $E_1,\ldots,E_k$, set $\widetilde E_i=V^TE_iV$.

Higher-order differentiability conjecture. The spectral function $f\circ\lambda$ is $k$ times continuously differentiable at $X$ if and only if $f$ is $k$ times continuously differentiable at the vector $\lambda(X)$. Moreover, there are $k$-tensor-valued maps

$$\mathcal A_\sigma:\mathbb{R}^n\longrightarrow T^{k,n},\qquad \sigma\in P^k,$$

such that

$$\nabla^k(f\circ\lambda)(X)[E_1,\ldots,E_k]=\sum_{\sigma\in P^k}\left\langle\mathcal A_\sigma(\lambda(X)),\widetilde E_1\circ_\sigma\cdots\circ_\sigma\widetilde E_k\right\rangle.$$

This conjecture describes the structure of higher-order derivatives of spectral functions; the equivalent tensorial formulation in the paper is a restatement of the same claim. The supplied text gives no evidence resolving it, so its status is left open.