Let $H$ be an infinite-dimensional separable $\mathbb R$-Hilbert space with orthonormal basis $(e_n)_{n\in\mathbb N}$, $$H_d:=\operatorname{span}\{e_1,\ldots,e_d\}\;\;\;\text{for }d\in\mathbb N$$ and $X$ be an $H$-valued random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$.
Is it possible that there exists a Borel measurable $p:H\to[0,\infty)$ such that $$\operatorname E\left[f(\pi_dX)\right]=\int\lambda^{\otimes d}({\rm d}x)p(\iota_dx)f(\iota_dx)\tag1$$ for all bounded Borel measurable $f:H_d\to\mathbb R$ and $d\in\mathbb N$, where $$\iota_d:\mathbb R^d\to H_d\;,\;\;\;a\mapsto\sum_{i=1}^da_ie_i$$ and $\pi_d$ is the canonical projection of $H$ onto $H_d$? The interpretation of $(1)$ is that every coordinate vector $$\left(\begin{matrix}\langle X,e_1\rangle_H\\\vdots\\\langle X,e_d\rangle_H\end{matrix}\right)$$ corresponding to the projection of $X$ onto the $H_d$ has a density $p_d$ with respect to the $d$-dimensional Lebesgue measure $\lambda^{\otimes d}$ and that all of them can be described by $p$ via $$p_d=p\circ\iota_d\tag2.$$ I wonder whether I'm missing something due to what it is impossible that such a $p$ can exist, since I haven't found anything in that direction in the literature.