A non-hereditarily-Lindelof regular space yields an ideal witness

By [F1], some subspace $Y\subseteq X$ has an open cover $\mathcal{V}$ with no countable subcover. Recursively for $\alpha <{\omega }_1$, the earlier chosen $V_{\xi }\in \mathcal{V}$ do not cover $Y$, so choose $x_{\alpha }\in Y\setminus {\bigcup }_{\xi <\alpha }{V}_{\xi }$ and then choose $V_{\alpha }\in \mathcal{V}$ containing $x_{\alpha }$. This also makes the $x_{\alpha }$ distinct.

Put $S=\{x_{\alpha }:\alpha <{\omega }_1\}$. If $\beta >\alpha$, construction gives $x_{\beta }\mathrm{\notin }{V}_{\alpha }$. Hence $V_{\alpha }\cap S$ is a neighbourhood of $x_{\alpha }$ contained in the initial segment $S_{\alpha }=\{x_{\xi }:\xi \le \alpha \}$. The union of these neighbourhoods for $\xi \le \alpha$ shows that every $S_{\alpha }$ is open in $S$, so $S$ is right-separated.

By [F2], $S$ is regular and Hausdorff. Apply [F3] inside $S$ to $x_{\alpha }\in S_{\alpha }$: choose open $U_{\alpha }$ with $x_{\alpha }\in U_{\alpha }\subseteq C_{\alpha }={\overline{U_{\alpha }}}^{,S}\subseteq S_{\alpha }$. The set $C_{\alpha }$ is closed in $S$ and countable because $\alpha <{\omega }_1$.

Let $\mathcal{I}$ be the ideal generated modulo finite by the $C_{\alpha }$. Explicitly, a countable $a\subseteq S$ lies in $\mathcal{I}$ exactly when $a{\subseteq }^{\ast }{\bigcup }_{\alpha \in u}{C}_{\alpha }$ for some finite $u\subseteq {\omega }_1$. The defining family consists of ${\omega }_1$ countable members, and the formula is downward closed, closed under finite unions, and contains every finite set.

Let $D\subseteq S$ be uncountable and inside $\mathcal{I}$, and let $E\subseteq D$ be countable. By inside-ness and step 4.1, $E\subseteq K={\bigcup }_{\alpha \in u}{C}_{\alpha }\cup F$ for some finite $u$ and finite $F\subseteq S$. The set $K$ is countable and closed in the Hausdorff space $S$: the $C_{\alpha }$ are closed and the finite set $F$ is closed. Choose $d\in D\setminus K$. Then the nonempty open subset $D\setminus K$ of $D$ misses $E$, so $E$ is not dense in $D$. Since this holds for every countable $E$, the space $D$ is nonseparable.

Let instead $D\subseteq S$ be uncountable and outside $\mathcal{I}$. For $x_{\alpha }\in D$, outside-ness gives $D\cap C_{\alpha }$ finite. Remove from $U_{\alpha }$ the finite closed set $(D\cap C_{\alpha })\setminus \{x_{\alpha }\}$. The result is an open neighbourhood in $S$ whose intersection with $D$ is exactly $\{x_{\alpha }\}$, so $D$ is discrete. Every dense subset of a discrete space is the whole space; because $D$ is uncountable, it is nonseparable.

Steps 1.1–4.1 give the promised right-separated sequence, closed neighbourhoods, and generated ideal, while steps 5.1 and 5.2 prove both nonseparability conclusions. The construction starts at $\alpha =0$ with no earlier cover members; every $S_{\alpha }$ and $C_{\alpha }$ is nonempty because it contains $x_{\alpha }$; finite generator lists and finite errors may be empty; and all nonempty recursive selections are the uses of AC recorded in [F5].