So the Ext groups of tmftC2 look like a bunch of ko's, and for degree reasons, its Adams spectral sequence must degenerate.
We claim that the gray classes are in the image of Extcs,t(k,H∗tmftC2), hence must be permanent. It suffices to prove that the generators under h0 and v14 are in the image, i.e. the classes in bidegree (8k−1,0). To do so, we note that they cannot be in the image of
The top dimensional cell in DΣ+∞RPn is always in degree 0. So the bigraded group ExtAs,t(k,H∗(tmfΣ+∞RP∞)) has a bottom vanishing line equal to that of ExtA(2)s,t(k,k). In particular, the corresponding generators at (8k−1,0) are all below this line, so are mapped injectively into ExtA(2)(k,H∗(P1)).
We now give a formal description of these classes. For any i∈Z, we let C(i) be the chart of Σ8i−1ko truncated to below the line y=−4x+6i−1. Then the gray classes are given by ⨁i≥1C(i) plus all its v28-multiples. We depict C(3) in Figure 2 for reference.