[Copied and extended from comments.]

First, it needs to be clarified in relation to the question that $\mathbb{R}$ does **not** satisfy the cohomological condition (viꝫ., having cohomological dimension $\leq 2$) in the hypothesis of the cited conjecture by Serre. Indeed, $\mathbb{R}$ has absolute Galois group cyclic of order $2$, and finite cyclic groups have periodic cohomology (so not at all finite cohomological dimension). So the following negative answer is not a counterexample to Serre's conjecture (😅).

Now a negative answer to the question posted is provided, for example, by the simply-connected semisimple algebraic group $G_2$ (the exceptional group of dimension $14$). In this case, the nonabelian Galois cohomology set $H^1(k, G_2)$ classifies octonion algebras over $k$ (Serre, *Galois Cohomology*, III, appendix 2, 3.3), as is to be expected since $G_2$ is the automorphism group of the octonions; but there are (exactly) two octonion algebras over the real numbers: the split octonions (which are the distinguished element of $H^1(\mathbb{R}, G_2)$) and the usual octonions. (Equivalently, since $G_2$ is also adjoint, $H^1(\mathbb{R}, G_2)$ classifies real forms of $G_2$ itself, of which there are exactly two: the split form, which is the automorphisms of the split octonions, and the compact form, which is the automorphisms of the usual octonions.) So $H^1(\mathbb{R}, G_2)$ has two elements: it is not trivial.

For more about the description of the Galois cohomology of semisimple groups over the reals, I recommend papers and talks by Mikhail Borovoi such as Borovoi & Timashev, Galois cohomology of real semisimple groups. For more about the Galois cohomology of $G_2$ and its relation to octonion algebras, see Springer & Veldkamp, *Octonions, Jordan Algebras and Exceptional Groups* (2000).

notsatisfy the condition of having cohomological dimension $≤2$ (since its Galois group is finite cyclic nontrivial, and such have periodic cohomology), and ⓑ if $G$ is, for example, the simply-connected semisimple algebraic group $G_2$, then $H^1(k,G_2)$ classifies octonion algebras over $k$ (Serre,Galois Cohomology, III, appendix 2, 3.3), and for $k=\mathbb{R}$ there are two of them (the split octonions and usual octonions), so $H^1(\mathbb{R},G_2)$ is nontrivial. $\endgroup$