Quadratic pants decomposition conjecture for bilipschitz embedded surfaces

Let $K$ be a surface of genus $g$ embedded in ${\mathbb{R}}^N$ by a bilipschitz map, and let $\gamma_1,\dots,\gamma_{3g-3}$ be the boundary curves of a pants decomposition of $K$. Here $\operatorname{length}\gamma_i$ denotes the length of $\gamma_i$, and $\operatorname{area} K$ denotes the area of $K$. Quadratic pants decomposition conjecture. There is a pants decomposition such that

$$\sum_i (\operatorname{length} \gamma_i)^2\lesssim \operatorname{area} K.$$

The implicit constant is independent of the genus. Without the embedding hypothesis, the claim is false for generic hyperbolic surfaces, while with genus-dependent constants it follows from the systolic inequality for closed surfaces. A surface formed by adding handles to a cube is presented as a potential counterexample.