Standard consumption CAPM applies a constant discount rate across all stocks, but surely that is odd if different companies face different costs of capital, as indeed they do. Take the companies with a higher cost of capital — in equilibrium they also should have higher rates of return as an offset. And those are (usually) the small stocks, and indeed we know there is a small stock premium (sometimes better expressed as a lower market to book premium) in the finance literature.
But that premium comes from the supply side arbitrage conditions, not from some odd properties of portfolio risk.
You will note that “the investment CAPM says that controlling for a few characteristics is sufficient to explain the cross section of expected returns.” Theory advocates claim that investment CAPM indeed passes that test: “…most anomalies turn out to be different manifestations of the investment and profitability effects.”