Conditions for traceability under a bound on the size of even-distance sets

We make partial progress towards a proof of Conjecture 189 of Written on the Wall II by showing that a connected graph $G$ satisfying $\max \{\mathrm{d}\mathrm{i}\mathrm{s}{\mathrm{t}}_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}(v):v\in V(G)\}\le d_2+1$, where $\mathrm{d}\mathrm{i}\mathrm{s}{\mathrm{t}}_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}(v)$ is the number of vertices at an even distance from $v$ and $d_2$ is the second smallest degree of $G$, is traceable whenever at least one of four conditions holds. These conditions involve the vertex-connectivity, order, and diameter of $G$.