Strict dead end elements in free soluble groups

Let $G$ be a group generated by a finite set $A$. An element $g\in G$ is a strict dead end of depth $k$ (with respect to $A$) if $\mathrm{\mid }g\mathrm{\mid }>\mathrm{\mid }g{a}_1\mathrm{\mid }>\mathrm{\mid }g{a}_1{a}_2\mathrm{\mid }>\mathrm{.}\mathrm{.}\mathrm{.}>\mathrm{\mid }g{a}_1{a}_2\mathrm{.}\mathrm{.}\mathrm{.}{a}_k\mathrm{\mid }$ for any $a_1,a_2, …, a_k\in A^{\pm1}$suchthattheword$a_1a_2… a_k$isfreelyirreducible\mathrm{.}(Here$|g|$isthedistancefrom$g$totheidentityintheCayleygraphof$G$.) We show that in finitely generated free soluble groups of degree $d\ge 2$ there exist strict dead elements of depth $k=k(d)$, which grows exponentially with respect to $d$.