The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ring, we get the Bass-Quillen conjecture:

Let $R$ be a regular ring and $P$ a projective module over $R[x_1,\dots, x_n]$, then $P \cong Q \otimes_R R[x_1,\dots, x_n]$ for a projective $R$-module $Q$.

This has been proven in many cases, for example for $R$ of Krull dimension $\leq 2$ or if $R$ is a localization of an affine $k$-algebra for $k$ a field.

Now one could put forward similar conjectures replacing polynomial variables by power or Laurent series variables:

(Power) Let $R$ be a regular ring and $P$ a projective module over $R[[x_1,\dots, x_n]]$, then $P \cong Q \otimes_R R[[x_1,\dots, x_n]]$ for a projective $R$-module $Q$.

(Laurent) Let $R$ be a regular ring and $P$ a projective module over $R((x_1,\dots, x_n))$, then $P \cong Q \otimes_R R((x_1,\dots, x_n))$ for a projective $R$-module $Q$.

If I understand the (affine) Horrocks Theorem correctly, then the Laurent series version of the conjecture actually implies the original Bass-Serre conjecture for a ring $R$ if $n=1$. Actually, Horrocks proves the Laurent series version for $R$ regular local either of dimension $\leq 1$ or of dimension $\leq 2$ and containing a field already in 1964. My question is now:

What is known about the power and Laurent series versions of the Bass-Quillen conjecture?