Let $D^n$ be the $n$-dimensional unit disk in euclidean $\mathbb{R}^n$. Larry Guth's Sponge Problem asks: Does there exist a constant $\epsilon=\epsilon_n$ such that every open subset $U\subset \mathbb{R}^n$ satisfying $vol(U)< \epsilon_n$ admits an expanding embedding $f: U\hookrightarrow D^n$?

Recall $f$ is expanding embedding iff the symmetric matrix ${}^tDf\cdot Df$ has all eigenvalues $\geq 1$. Equivalently iff $||D_xf(v)||\geq ||v||$ for every $x\in U$ and tangent vector $v\in T_x U$. Equivalently iff $f$ increases the length of all curves in $U$.

My best estimate $\epsilon_n^*$ for $\epsilon_n$ comes from obvious example of two disks of radius $1/2$ kissing in $D$, which disks cannot be expanded embedded into $D$. Thus I estimate $$\epsilon^*_2=2 (\pi (1/2)^2)=\pi^2/4 ,$$ and more generally $$\epsilon_n^*=2 \cdot \text{Vol}(D(0,1/2))=\frac{\pi^{n/2}}{2^{n-1} \cdot\Gamma(n/2+1)} ,$$ where $D(0,1/2)$ is the $n$-ball of radius $1/2$.

I've been interested in Sponge problem for a few years, and for all that time I've yet to improve on the estimate $\epsilon^*_n = \epsilon_n$. No matter how "spongy" or ``swiss cheesed" the open set $U$ is, only the existence of too many large disjoint disks appears to be obstruction to being expanded-embedded into $D$.

Question: Is anybody aware of any progress in Guth's Sponge Problem, or candidate open sets $U\subset \mathbb{R}^n$ with $vol(U)<\epsilon_n^*$ which cannot be expanded embedded into $D$?