PeriodicInterpolation does not work in ElementMeshInterpolation

I want to use ElementMeshInterpolation to generate interpolation function with periodic boundary condition.

I use below data as an example

data=Flatten[Table[{i,j,Sin[i+j]},{i,0,2\[Pi],2\[Pi]/50},{j,0,2\[Pi],2\[Pi]/50}],1]; ListContourPlot[data] 

which gives

enter image description here

This data is periodic along x and y direction.

Using Interpolation

f = Interpolation[data, PeriodicInterpolation -> True]; {ContourPlot[f[x, y], {x, 0, 2 \[Pi]}, {y, 0, 2 \[Pi]}],   ContourPlot[f[x, y], {x, 0, 4 \[Pi]}, {y, 0, 4 \[Pi]}]} 

gives

enter image description here

We can see the Interpolation function is fine with periodic condition as wanted.

using ElementMeshInterpolation

Though Interpolation works fine for this data set. But Interpolation has problem that it frequently run into "femimq" problem. So ElementMeshInterpolation on a refined mesh is necessary sometimes.

mesh = ToElementMesh[data[[;; , 1 ;; 2]]]; f = ElementMeshInterpolation[{mesh}, data[[;; , -1]],     PeriodicInterpolation -> {True, True}]; {ContourPlot[f[x, y], {x, 0, 2 \[Pi]}, {y, 0, 2 \[Pi]}],   ContourPlot[f[x, y], {x, 0, 4 \[Pi]}, {y, 0, 4 \[Pi]}]} 

this gives

enter image description here

You see the generated Interpolation function has no periodicity.

using ListInterpolation

mesh can also be used in ListInterpolation

mesh = ToElementMesh[data[[;; , 1 ;; 2]]]; f = ListInterpolation[data[[;; , -1]], mesh,     PeriodicInterpolation -> {True, True}]; {ContourPlot[f[x, y], {x, 0, 2 \[Pi]}, {y, 0, 2 \[Pi]}],   ContourPlot[f[x, y], {x, 0, 4 \[Pi]}, {y, 0, 4 \[Pi]}]} 

but this gives the same result as ElementMeshInterpolation.

So the question is how to correctly make periodic interpolation function using ElementMeshInterpolation.