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Earlier quoted context omitted.
It couldn't.
Pretty damn close though! I haven't seen an explanation of what the second figure is trying to show so I'm not sure about that one. (And also their assertion that no further explanation is necessary is clearly bullshit.)
The second figure is actually showing another arrangement of n²+2 small unit equilateral triangles covering an equilateral triangle of side length n+ε.
Earlier quoted context omitted.
The paper doesn't answer the problem in its title (n²+1) but demonstrates two different "advancements" towards it (n²+2) > We have posed a fine (in our opinion) open problem and reported two distinct “behold-style” proofs of our advance on this problem. There's also a linked PDF which, if I'm reading it correctly, trivially proves n²+1 is impossible.
No, n^2+1 is an open problem (conjectured to be impossible though)