Conjecture about cardinality of Lindelöf spaces

The conjecture asks for a Lindelöf space where all singletons are G_δ sets and which has cardinality > 𝔠.

This is Problem 1 in https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf

Reference:

• Selected Old Open Problems in General Topology by A. V. Arhangel’skii

theorem CardinalityLindelof.HasGδSingletons.lindelof_card : ∃ (X : Type) (x : TopologicalSpace X), HasGδSingletons X ∧ LindelofSpace X ∧ Cardinal.continuum < Cardinal.mk X

Is there a Lindelöf space with singletons as Gδ sets with cardinality greater than the continuum?