Is the following correct and/or a (simple) known result?
Let $X$ be a local martingale and $H$ an integrand for $X$, such that the stochastic integral $\int H\cdot dX\ge x$ for some random variable. Then $\int H\cdot dX$ is also a local martingale.
I would prove it in analogy to the proof in Corollaire 3.5 of this paper:
Define the process $V_t=\int^t_0 H\cdot dX-x\ge 0$ and the sequence of stopping times $T_n=\inf\{t:V_t\ge n\}$. Thus, $(V^{T_n})_{t-}\le n$ which together with $V_t\ge 0$ yields $\Delta V^{T_n}=\Delta\left(\int H\cdot d(X^{T_n})\right)\ge -n$. The assertion then follows from Proposition 3.3 of the same paper.
Proposition 3.3: The stochastic integral $\int H\cdot dX$ is a local martingale if and only if there exists a sequence of stopping times with limit $\infty$ and a series of integrable negative random variables $\theta_n$, such that $ H\cdot\Delta X^{T_n}\ge\theta_n$.