The Genius conjecture on linearity of the functions $F_i$

Let $p$ be a fixed positive integer. Let $u_1=1$, let $u_2,\dots,u_p$ and $d_1,\dots,d_p$ be variables, and let $F_1=1$ and $F_i=F_i(\{u_k\},\{d_k\})$ for $i=2,\dots,p$ be the unique functions satisfying

$$[x^p]e^{\sum_{i=1}^p (yu_i+d_i)x^i}=[x^p]e^{\sum_{i=1}^p yF_i x^i},$$

where $[x^s]f$ denotes the coefficient of $x^s$ in $f$ and $[x^p]e^{\sum_{i=1}^p d_i x^i}=0$. The Genius linearity conjecture. For each $i=2,\dots,p$, $F_i(\{u_k\},\{d_k\})$ is linear in its dependence on the variables $\{u_k\}$. This is one of the paper's principal conjectures about the functions arising from the coefficient identity; the supplied text gives no resolution.