Linear realizations for small $k$

Let $L$ be a multiset of linear lengths, where $\{1^a,(y-k)^b,y^c\}$ denotes the multiset containing $a$, $b$, and $c$ copies of $1$, $y-k$, and $y$, respectively. A multiset is linearly realizable if it is the multiset of linear lengths of a Hamiltonian path in a complete graph whose vertices are labelled consecutively. Linear-realization conjecture for small $k$. When

$$k<y/2,$$

multisets of the form

$$\{1^a,(y-k)^b,y^c\}$$

are linearly realizable for all $a\geq y$. The conjecture is presented as a possible next extension of the paper's methods and as a step toward proving the Coprime BHR conjecture for supports of size three; the source does not state that it has been resolved.