The injectivity conjecture for the map $\bar{\varphi}$

Fix non-negative integers $d_1,d_2$, let $M_{d_1,d_2}$ be the corresponding quotient space of alternating bihomogeneous polynomials, and let

$$\bar{\varphi}:M_{d_1,d_2}\longrightarrow \mathbb C[\rho_1,\rho_2,\ldots]_k$$

be the induced linear map, where $k={n\choose 2}-d_1-d_2$. Injectivity conjecture. The linear map $\bar{\varphi}$ is injective. The preceding results establish injectivity when $k\leq n-3$ (under the stated ordering $d_2\leq d_1$), while the computations reported in the source support the conjecture in the more general range.