The k-sum embedding conjecture for graph families

For a graph family $\mathcal F$, let $\oplus_k\mathcal F$ denote its closure under $k$-clique sums, and let $c_1(\mathcal F)$ denote the supremum of the least distortion needed to embed members of $\mathcal F$ into $L_1$.

$k$-sum embedding conjecture. For any family of graphs $\mathcal F$, we have

$$c_1(\mathcal F)=O(1) \quad\text{if and only if}\quad c_1(\oplus_k\mathcal F)=O(1) \text{ for every } k\in\mathbb N.$$

Lee and Sidiropoulos showed that this conjecture together with the planar embedding conjecture is equivalent to the GNRS conjecture. The case $k=1$ is folklore, and the case $k=2$ has seen progress but remains open.