Equal matroids conjecture for CFN join varieties of a pair of tree pairs

Let $T_1,T_2,S_1,S_2$ be the four-leaf trees specified by the paper's Equation (badTreeCase), and let $V_{T_1} \ast V_{T_2}$ and $V_{S_1} \ast V_{S_2}$ be their associated CFN join varieties. For an algebraic variety $V$, write $\mathcal{M}(V)$ for its associated algebraic matroid.

Equal matroids conjecture. The two join varieties have the same algebraic matroid:

$$\mathcal{M}(V_{T_1} \ast V_{T_2})=\mathcal{M}(V_{S_1} \ast V_{S_2}).$$

The authors propose this because their matroid-based algorithm found no certificate distinguishing this pair, although the corresponding phylogenetic-invariant ideals are known to be different. If true, this would provide an instance in which distinct ideals define the same matroid; the supplied text does not report a resolution.