Let \(X\) and \(Y\) be Banach spaces. Let \({\mathcal{L}}(X,Y)\) denote the space of bounded linear operators from \(X\) to \(Y\). Let \({\mathcal{F}}(X,Y)\), \({\mathcal{K}}(X,Y)\), and \({\mathcal{W}}(X,Y)\) be its subspaces of finite-rank, compact, and weakly compact operators, respectively. Finally, let \(\tau_c\) denote the compact-open topology on \({\mathcal{L}}(X,Y)\). A Banach space \(X\) has the strong approximation property (strong AP) if, for every separable reflexive \(Y\) and every \(R\in{\mathcal{K}}(X,Y)\), one has \(R\in\overline{ \{ U\in{\mathcal{F}}(X,Y) : \| U\|\leq\lambda_R\}}^{\tau_c}\) for some \(\lambda_R>0\). The strong AP was introduced and studied in [\textit{E. Oja}, J. Math. Anal. Appl. 338, No. 1, 407--415 (2008; Zbl 1148.46015)], where it was proved that \(X\) has the strong AP if and only if, for every \(Y\) and every \(R\in{\mathcal{K}}(X,Y)\), one has \(R\in\overline{\{RS : S\in{\mathcal{F}}(X,X) : \| RS\|\leq\lambda_R\}}^{\tau_c}\) for some \(\lambda_R>0\). The latter form is suitable for extending the notion by replacing \({\mathcal{F}}(X,X)\) with a convex subset \({\mathcal{A}}(X,X)\) of \({\mathcal{L}}(X,X)\), see [\textit{A. Lissitsin}, Stud. Math. 211, No. 3, 199--214 (2012; Zbl 1275.46009)]. The authors make a ``cosmetic'' generalization by saying, for a given \(T\in{\mathcal{L}}(X,X)\), that \(X\) has the strong \(T\)-\({\mathcal{A}}\)-approximation property (strong \textit{T-}\({\mathcal{A}}\)-AP) if for every \(Y\) and every \(R\in{\mathcal{K}}(X,Y)\), one has \(RT\in\overline{\{RS : S\in{\mathcal{A}}(X), : \| RS\|\leq\lambda_R\}}^{\tau_c}\) for some \(\lambda_R>0\). Hence the strong \({\mathcal{A}}\)-AP is the strong \(T\)-\({\mathcal{A}}\)-AP for \(T=\mathrm{id}_X\). For \(\lambda\geq 1\), a Banach space \(X\) has the weak \(\lambda\)-bounded approximation property (weak \(\lambda\)-BAP) if for every \(Y\) and every \(R\in{\mathcal{W}}(X,Y)\) one has \(\mathrm{id}_X\in\overline{\{S\in{\mathcal{F}}(X,X) : \| RS\|\leq\lambda\|R\|\}}^{\tau_c}\). The weak \(\lambda\)-BAP was introduced and studied in [\textit{Å. Lima} and \textit{E. Oja}, Math. Ann. 333, No. 3, 471--484 (2005; Zbl 1097.46012)], and its convex version (where \({\mathcal{F}}(X,X)\) is replaced with \({\mathcal{A}}(X,X)\)) in a number of papers. The similar ``cosmetic'' generalization is proposed by the authors, calling it the \(T\)-\({\mathcal{A}}\)-\(\lambda\)-bounded approximation property (weak \(T\)-\({\mathcal{A}}\)-\(\lambda\)-BAP) of \(X\). The strong AP lies between the weak BAP and the AP, being strictly stronger than the AP. Oja [loc.\,cit.]\ conjectured that the strong AP and the weak BAP are different properties. The main result in the paper under review is Theorem 1.3 which disproves this conjecture by showing that if \({\mathcal{A}}(X,X)\) is a convex subset of \({\mathcal{K}}(X,X)\) and \(T\in{\mathcal{L}}(X,X)\), then \(X\) has the weak \(T\)-\({\mathcal{A}}\)-BAP whenever \(X\) has the strong \(T\)-\({\mathcal{A}}\)-AP. This allows, basing on known examples, to indicate further examples of spaces having the AP but failing the strong AP, e.g., certain subspaces of \(c_0\) and \(\ell_1\). The proof of Theorem 1.3 relies on Theorem 1.1, and in the Corrigendum [Zbl 1334.46016] on Proposition 2.1, which are both versions of the ``Radon-Nikodým property impact theorem'' from [\textit{E. Oja}, RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 100, No. 1--2, 325--331 (2006; Zbl 1112.46017)]. It also relies on characterizations of the strong and weak \(T\)-\({\mathcal{A}}\)-BAPs which are similar (concerning also the proofs) to those in Theorem 2.4 of the Lima-Oja paper [loc. cit.]. Reviewer's remark: Characterizations of the strong and weak \(\mathrm{id}_X\)-\({\mathcal{A}}\)-BAPs have already appeared in Proposition 5.2 and Corollary 5.4 of Lissitsin's [loc.\,cit.]\ unified investigation of the strong and weak BAPs.