Secure-torsor counting conjecture for finite étale group schemes

Let $F$ be a global field, let $G$ be a non-trivial finite étale tame $F$-group scheme, and let $H:BG(F)\to\mathbb R_{>0}$ be a height. A torsor $x\in BG(F)$ is secure if it is not in the image of any breaking thin map $BR(F)\to BG(F)$, where such maps arise from inner twists of $G$ and closed subgroup schemes $R$ and have pullback height with lexicographically larger $(a,b)$-invariants.

Secure-torsor counting conjecture. If $F$ is a number field, there exists $C>0$ such that

$$\#\{x\in BG(F)\mid x\text{ is secure and }H(x)\leq B\}\sim_{B\to\infty}CB^{a(H)}\log(B)^{b(H)-1}.$$

If $F$ is a function field, then

$$\#\{x\in BG(F)\mid x\text{ is secure and }H(x)\leq B\}\asymp_{B\to\infty}B^{a(H)}\log(B)^{b(H)-1}.$$

This is proposed as an analogue of Malle's conjecture for torsors of finite group schemes. The provided text gives no resolution status.