Recall that the birthday $B(G)$ of a combinatorial game $G$ is recursively defined as the least ordinal $\alpha$ such that $G = \{L | R\}$ for some sets of games $L$, $R$ with birthdays less than $\alpha$.

It is straightforwardly shown through Conway induction that $B(G + H) \le B(G) \oplus B(H)$ where $\oplus$ is the natural sum. If $G$ and $H$ are ordinals, the equality is attained. As a corollary, the set of games with birthday less than some $\omega^\alpha$ forms a subgroup under addition.

I'm interested in a similar result for surreal multiplication. The obvious guess is $B(G \times H) \le B(G) \otimes B(H)$, where $\otimes$ is the natural product. However, I've completely failed at proving this or finding a reference, besides the easy case where $G$ and $H$ are dyadic rationals (in which case a simple characterization for their birthday can be employed).

The issue is that a priori, the games $G^LH + GH^L - G^LH^L$, etc. can have a birthday of at least $3B(G^LH^L)$. My guess is that whenever $G$ and $H$ have "nice" options (with smaller birthdays, etc.), the games $G^LH + GH^L - G^LH^L$, etc. can be rewritten in some other form that reveals a smaller birthday. However, after trying for various hours, I've come up blank.