Is it still open to determine the complexity of computing the treewidth of planar graphs? For a constant $k \in \mathbb{N}$, one can determine in linear time, given an input graph $G$, whether its treewidth is $\leq k$. However, when both $k$ and $G$ are given as input, the problem is NP-hard. (Source).

However, when the input graph is planar, much less seems to be known about the complexity. The problem was apparently open in 2010, a claim that also appeared in this survey in 2007 and on the Wikipedia page for branch decompositions. Conversely, the problem is claimed NP-hard (without proof of reference) in an earlier version of the previously mentioned survey, but I assume this is an error.

Is it still open to determine the complexity of the problem, given $k \in \mathbb{N}$ and a planar graph $G$, of determining $G$ has treewidth $\leq k$? If it is, was this claimed in a recent paper? Are any partial results known? If it isn't, who solved it?
As far as I know the NP-completeness of computing the treewidth of a planar graph is still open. The most recent reference I know is a survey by Bodlaender from 2012 called `Fixed-Parameter Tractability of Treewidth and Pathwidth' that appeared in the festschrift for Mike Fellows' 65th birthday. The problem is listed in the conclusion of the survey.