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 $D$). Remarks of @Balarka Sen in comments below yields the following slightly volume decreasing improvement. Let $A$ be an Apollonian packing of $U'$ which has minimal volume. (Recall $A$ is some infinite maximal disk packing of $U'$). Then we estimate $$\epsilon^*_n=\inf_{A} vol(A),$$ where the infimum is taken over all Apollonian packings $A$ of $U'$. 

Obviously $\epsilon_n^*$ is *slightly smaller* than the volume of $U'$, which is $1/2^{n-1}$ times the volume of the unit disk. Replacing $U'$ by an Apollonian packing $A$ is interesting idea, but does not dramatically reduce the volume. 


**Question**: Is anybody aware of any further 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 $D$?