Procedural realistic terrain for hex tile map - what's causing the edges to stick up?

I'm working on procedurally generating realistic terrain on hex tiles for a strategy game. I've got it working nicely except where a tile is concave the edge is still visible as a one-sided face that sticks up. I think it might have something to do with how Unity draws triangles. Any ideas?

The tile changes shape correctly except the edges seem to stick up like this: https://i.imgur.com/NGsDmSl.png

Here's what the mesh looks like in Blender: https://i.imgur.com/TL6y2jj.png

Here's my code:

 void MorphTerrain()
 {

     Vector3[] verts = GetVertices();
     Vector3[] newVerts = new Vector3[verts.Length];

     float bottomLayer = GetBottomLayer(verts);

     for (int i = 0; i < verts.Length; i++)
     {
         if (verts[i].z > bottomLayer) //Because blender, z is the axis that will control height
         {
             newVerts[i] = verts[i];
             continue;
         }
         float sampleX = -(transform.position.x + (verts[i].x * 100));
         float sampleY = transform.position.z - (verts[i].y * 100);

         newVerts[i] = new Vector3(verts[i].x, verts[i].y, GetNoiseValue(sampleX, sampleY));
     }

     SetVertices(newVerts);
 }
 Vector3[] GetVertices()
 {
     return mesh.vertices;
 }

 void SetVertices(Vector3[] verts)
 {
     mesh.vertices = verts;
     mesh.RecalculateBounds();
 }

 float GetNoiseValue(float x, float y)
 {
     float xCoord = (x + xOffset) * noiseResolution;
     float yCoord = (y + yOffset) * noiseResolution;

     float rise = height * Mathf.PerlinNoise(xCoord, yCoord);
     return rise;
 }

 private float GetBottomLayer(Vector3[] verts)
 {
     float result = 1f;
     foreach (Vector3 v in verts)
     {
         if (v.z < result) result = v.z;
     }
     return result;
 }

Thanks in advance for your help!