Regularity conjecture for cycles of reduced forms of discriminant $a^2+4$

Let a cycle of reduced forms have caliber $l$, discriminant $a^2+4$, and sum $n$. Regularity conjecture. There is an integer $r\geq 0$ such that

$$n=(2r+1)l+1.$$

If $r>0$, the cycle consists of imprimitive forms, meaning that the greatest common divisor of the coefficients of each form is greater than $1$. These conjectured regularities arise from computational evidence concerning cycles with fixed sum; the source does not provide a proof or resolution.