On $\Sigma$-rigid presentations of the real order

For arbitrary tuples of real parameters $\bar p$, we prove the existence and effective infiniteness of the class of the linear orders on $\mathbb R$ of type $\langle\mathbb R,<\rangle$ which are $\Sigma$-definable over $\mathbb{HF(R)}$ with parameters $\bar p$ and have no nontrivial $\Sigma$-definable self-embeddings with parameters $\bar p$.