Discriminator conjecture for half-integer quadratic sequences with coefficient ratio divisible by 3

Let $b$ and $c$ be non-zero integers such that $3\nmid bc$. For a sequence

$$(\operatorname{qr}(n))_{n\geq 0}=\left(\frac{3c}{2}n^2+\frac{bc}{2}n\right)_{n\geq 0},$$

let $D_{\operatorname{qr}}(n)$ denote its discriminator, the least modulus for which the first $n$ sequence values are pairwise distinct. The discriminator conjecture. For every such sequence, there exists at least one value of $n\geq 1$ such that

$$D_{\operatorname{qr}}(n)\ne 3^{\lceil\log_3 n\rceil}.$$

The preceding result rules out the claimed discriminator formula in most cases of half-integer quadratic coefficients; this conjecture concerns the remaining case in which the quadratic coefficient is $3c/2$, with $3\nmid c$ and $c\mid\beta'$ as represented by the linear coefficient $bc/2$. The authors report that they were unable to find any examples satisfying the formula, but the conjecture remains open.