The PBW decomposition conjecture for the current algebra

Let $\mathcal A_q$ be the current algebra over $\mathbb F$, with generators $\mathcal W_{-k}$, $\mathcal W_{k+1}$, $\mathcal G_{k+1}$, and $\mathcal{\tilde G}_{k+1}$ for $k\in\mathbb N$. Let $\lambda_1,\lambda_2,\ldots$ be mutually commuting indeterminates, let $\mathbb F[\lambda_1,\lambda_2,\ldots]$ be their polynomial algebra, and for $Y\subseteq\mathcal A_q$ let $\langle Y\rangle$ denote the subalgebra generated by $Y$. PBW decomposition conjecture. The following hold: (i) $\mathbb F[\lambda_1,\lambda_2,\ldots]\to\langle\mathcal W_0,\mathcal W_{-1},\ldots\rangle$, $\lambda_{k+1}\mapsto\mathcal W_{-k}$, is an algebra isomorphism; (ii) $\mathbb F[\lambda_1,\lambda_2,\ldots]\to\langle\mathcal W_1,\mathcal W_2,\ldots\rangle$, $\lambda_{k+1}\mapsto\mathcal W_{k+1}$, is an algebra isomorphism; (iii) $\mathbb F[\lambda_1,\lambda_2,\ldots]\to\langle\mathcal G_1,\mathcal G_2,\ldots\rangle$, $\lambda_{k+1}\mapsto\mathcal G_{k+1}$, is an algebra isomorphism; (iv) $\mathbb F[\lambda_1,\lambda_2,\ldots]\to\langle\mathcal{\tilde G}_1,\mathcal{\tilde G}_2,\ldots\rangle$, $\lambda_{k+1}\mapsto\mathcal{\tilde G}_{k+1}$, is an algebra isomorphism; and (v) the multiplication map

$$\langle\mathcal W_0,\mathcal W_{-1},\ldots\rangle\otimes\langle\mathcal G_1,\mathcal G_2,\ldots\rangle\otimes\langle\mathcal{\tilde G}_1,\mathcal{\tilde G}_2,\ldots\rangle\otimes\langle\mathcal W_1,\mathcal W_2,\ldots\rangle\to\mathcal A_q,$$

$u\otimes v\otimes w\otimes x\mapsto uvwx$, is an isomorphism of vector spaces.

A proof would yield a PBW basis for $\mathcal A_q$. The statement is presented as a variation on an earlier conjecture, and no resolution is supplied.