Smurov's plank covering conjecture for planar convex bodies

Suppose a collection of planks in the plane has total width $1$, and let $B$ be a convex body with perimeter at most $2$.

Smurov's conjecture. The planks can be translated so that their translates cover $B$.

This is the second conjecture attributed to Mikhail Smurov in the source. The paper presents it as open and gives no further partial result for this formulation.