Colinearity in a two-households-model due to similar euler equations

No, I am saying that there might be something like the permanent income hypothesis at work. Any shock that moves the debt position will move it permanently and there is no tendency to return to the old level. That is a common thing happening under perfect foresight.
Things usually change under stochastics. For example, the unit root in small open economy models vanishes when agents take future shocks into account. The same happens in consumption Euler equations due to Jensen’s Inequality.