Symmetric adjoining conjecture for tensor rank

Let $T\in\mathbb{C}^{I\times I\times I}$ be a symmetric tensor, and let $\mathcal{M}\subset\mathbb{C}^{I\times I}$ be a set of symmetric rank-one matrices. Let $\mathcal{T}$ be obtained from $T$ by the symmetrical adjoining of the matrices in $\mathcal{M}$, and let $\mathcal{E}_{\operatorname{span}\mathcal{M}}(T)$ denote the associated family of symmetric tensors obtained by adjoining the span of $\mathcal{M}$. Symmetric adjoining conjecture. The symmetric rank of $\mathcal{T}$ equals

$$3\dim\operatorname{span}\mathcal{M}$$

plus the minimal symmetric rank of a symmetric tensor in $\mathcal{E}_{\operatorname{span}\mathcal{M}}(T)$. The author could not prove this even when $\mathcal{M}$ consists of one matrix, and the paper uses it as a conjectural symmetric counterpart of a rank formula for ordinary adjoining of slices.