Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality

Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality References Is there a linearly Lindelöf space $X$ with $cf(L(X))> \\aleph_{0}$ (where $L$ is the cardinal function Lindelöf degree)? $L(X)$ must be a limit cardinal, like $\\aleph_{\\omega_{1}}$ or an inaccessible cardinal, because if $L(X)$ is a successor cardinal, then there is an open cover $C$ of size $L(X)$ with no subcover of smaller size, and such space cannot be linearly Lindelöf (a space is linearly Lindelöf iff every open cover $C$ has a subcover $S$ with $cf(|S|)= \\aleph_{0}$). But consider the following situation for a space $X$: let $\\kappa$ be a limit cardinal of uncountable cofinality (like $\\aleph_{\\omega_{1}}$), and $\\Gamma = \\{ \\lambda_{\\alpha} : \\alpha < cf(\\kappa) \\}$ a set of infinite cardinals, with $cf(\\lambda_{\\alpha}) = \\aleph_{0}$ for every $\\alpha < cf(\\kappa)$, and $\\sup \\Gamma = \\kappa$; and for every $\\alpha < cf(\\kappa)$ there is an open cover $C_{\\alpha}$ with $|C_{\\alpha}| = \\lambda_{\\alpha}$ and no subcover of smaller size. This space $X$ may be linearly Lindelöf with $L(X) = \\sup \\Gamma = \\kappa$.