It is proved that for a –dimensional compact metrizable space the infinite real projective space is an absolute extensor of if and only if the real projective plane is an absolute extensor of (see Theorems 1.2 and 1.5).
"Extensions of maps to the projective plane." Algebr. Geom. Topol. 5 (4) 1711 - 1718, 2005. https://doi.org/10.2140/agt.2005.5.1711