Refined separation-number conjecture for the complete graph K4

Let $\mathsf{sep}(K_4,a,b)$ denote the separation threshold for the complete graph $K_4$.

K4 separation-number conjecture.

$$\mathsf{sep}(K_4,a,b)=\left\{\begin{array}{ll} \left\lfloor\frac{2(a-b)}{3}\right\rfloor, & b\leq a<2b,\\ \left\lceil\frac{4a-6b-1}{3}\right\rceil, & 2b\leq a<3b,\\ 2a-4b, & 3b\leq a<4b,\na, & a\geq4b. \end{array}\right.$$

The paper states that only the range $2b\leq a<3b$ remained to be verified. Thus the displayed piecewise formula is proposed as a refinement of the general separation-number conjecture, with the middle case identified as the unresolved part in the supplied text.