Summary: A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph \(G\), the proper connection number \(\operatorname{pc}(G)\) of \(G\) is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of \(G\) is connected by at least one proper path in \(G\). In this paper, we consider two conjectures on the proper connection number of graphs. The first conjecture states that if \(G\) is a noncomplete graph with connectivity \(\kappa(G) = 2\) and minimum degree \(\delta(G) \geq 3\), then \(\operatorname{pc}(G) = 2\), posed by V. Borozan et al. [ibid. 312, No. 17, 2550–2560 (2012; Zbl 1246.05090)]. We give a family of counterexamples to disprove this conjecture. However, from a result of C. Thomassen [in: Selected topics in graph theory. Vol. 3. London (UK) etc.: Academic Press Ltd. 97–131 (1988; Zbl 0659.05062)] it follows that 3-edge-connected noncomplete graphs have proper connection number 2. Using this result, we can prove that if \(G\) is a 2-connected noncomplete graph with \(\operatorname{diam}(G) = 3\), then \(\operatorname{pc}(G) = 2\), which solves the second conjecture we want to mention, posed by X. Li and C. Magnant [“Properly colored notions of connectivity – a dynamic survey”, Theory Appl. Graphs 0, No. 1, Article 2 (2015)].