Applied time series research often faces the challenge that (a) potentially relevant variables are unobservable, (b) it is fundamentally uncertain which covariates are relevant. Thus cointegration is often analyzed in partial systems, ignoring potential (stationary) covariates. By simulating hypothesized larger systems Benati (2015) found that a nominally significant cointegration outcome using a bootstrapped rank test (Cavaliere, Rahbek, and Taylor, 2012) in the bivariate sub-system might be due to test size distortions. In this note we review this issue systematically. Apart from revisiting the partial-system results we also investigate alternative bootstrap test approaches in the larger system. Throughout we follow the given application of a long-run Phillips curve (euro-area inflation and unemployment). The methods that include the covariates do not reject the null of no cointegration, but by simulation we find that they display very low power, such that the (bivariate) partial-system approach is still preferred. The size distortions of all approaches are only mild when a standard HP-filtered output gap measure is used among the covariates. The bivariate trace test p-value of 0.027 (heteroskedasticity-consistent wild bootstrap) therefore still suggests rejection of non-cointegration at the 5% but not at the 1% significance level. The earlier findings of considerable test size distortions can be replicated when instead an output gap measure with different longer-run developments is used. This detrimental effect of large borderline-stationary roots reflects an earlier insight from the literature (Cavaliere, Rahbek, and Taylor, 2015).