The question has a negative answer. The technique is essentially due to Jockusch and Posner.

Proof: Let $x$ be a real in which every constructible real is recursive. Now $$A=\{r\mid r\mbox{ is Martin-L\" of random relative to }x \wedge x\oplus r\geq_T x',\mbox{ the Turing jump of }x.\}$$ Then $A$ is null, $r\oplus x$ ranges over the upper cone $\geq_T x'$, and only contains $L$-random reals. Now for any $z\geq_T x'$ and $r \in A$, we have that

(1). $r\triangle x \in A$; and

(2). $(r\triangle x) \oplus r\equiv_T x\oplus r\geq_T x'$.

So the $L$-ideal generated by $A$ is $\mathbb{R}$.