Structural rank conjecture for the p-adic logarithm matrix

Fix an embedding $\iota_p:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_p$, let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ of dimension $d$, and let $\{\omega_1,\ldots,\omega_d\}$ be a $\overline{\mathbb{Q}}$-basis of $\Omega_A^1$. For $x_1,\ldots,x_r\in A(\overline{\mathbb{Q}})$ that are $\mathbb{Z}$-linearly independent, define

$$\mathcal{L}_{\underline{x}}=(\log_{\omega_i}(x_j))\in M_{d\times r}(\overline{\mathbb{Q}}_p),$$

where $\log_{\omega_i}$ is obtained by composing the $p$-adic logarithm of $A$ with the linear functional corresponding to $\omega_i$. The structural rank conjecture for $\mathcal{L}_{\underline{x}}$ asserts that its rank over $\overline{\mathbb{Q}}_p$ equals its structural rank, namely the rank obtained by replacing a $\mathbb{Q}$-basis of the entries by algebraically independent variables. This conjecture predicts the maximal rank allowed by the rational linear relations among the entries of the logarithm matrix. The source gives no resolution or further evidence for this conjecture.