= Solution
Totality of the <binary relation> is expressed by
$$
\boxed{\tau=\forall x\forall y\,(R(x,y)\lor R(y,x)).}
$$
When $R$ is already an order, this adds comparability and makes it a <total order>. In fact this <first-order sentence> alone entails $\rho=\forall xR(x,x)$.
By the definition of <semantic entailment>, $\tau\models\rho$ means that every <first-order structure> satisfying $\tau$ also satisfies $\rho$. Let $\mathcal M$ be any such <first-order structure>, and let $a$ be any element of its <domain of a first-order structure>. The instance of $\tau$ with $x=y=a$ gives $R(a,a)\lor R(a,a)$, which is true only if $R(a,a)$ is true. Since $a$ was arbitrary, $\mathcal M\models\forall xR(x,x)$. Therefore
$$
\boxed{\tau\models\rho.}
$$
There is no exception for $x=y$ in the displayed totality axiom; that case is exactly what forces the <reflexive relation> property.
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