Let $D^n$ be the $n$-dimensional unit radius 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 previous best estimate $\epsilon_n^*$ for $\epsilon_n$ comes from obvious example of two disks $D_1, D_2$ of radius $1/2$ kissing, which disks $U:=D_1\cup D_2$ cannot be properly expanded embedded into $D$. (The two kissing disks barely embed into $D$).

The remarks of @Balarka Sen in comments yield the following improvement. Let $A$ be an Apollonian packing of $U$ which has minimal volume. Then we estimate $\epsilon^*_n$ as the minimal volume of an Apollonian packing of two kissing disks of radius $1/2$. This yields $$\epsilon^*_n<2 \cdot \text{Vol}(D(0,1/2))=\frac{\pi^{n/2}}{2^{n-1} \cdot\Gamma(n/2+1)},$$ 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 $\epsilon^*_n = \epsilon_n$. (Although replacing $U$ with an Apollonian packing $A$ of minimal volume is new idea I learned from the comments).

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 $D$?