Strict lifts and cohomological obstructions in symplectic Lie--Rinehart--Jacobi prequantization

Let $(\G,ρ,ω)$ be a symplectic Lie--Rinehart--Jacobi algebra over a commutative algebra $A$, equivalently a Lie--Rinehart algebra $(\G,\barρ)$ endowed with a $1$-cocycle $χ$ and a nondegenerate $2$-form closed for the twisted differential $d_ρ=d_{\barρ}+χ\wedge$. Starting from the standard extension