Summary: The total vertex irregularity strength \(\mathrm{tvs}(G)\) of a simple graph \(G(V, E)\) is the smallest positive integer \(k\) so that there exists a function \(\varphi:V \cup E \rightarrow [1, k]\) provided that all vertex-weights are distinct, where a vertex-weight is the sum of labels of a vertex and all of its incident edges. In the paper [\textit{Nurdin} et al., Discrete Math. 310, No. 21, 3043--3048 (2010; Zbl 1208.05014)], two conjectures regarding the total vertex irregularity strength of trees and general graphs were posed as follows: (i) for every tree \(T\), \(\mathrm{tvs} (T) = \max\{\lceil(n_1+ 1)/2\rceil, \lceil(n_1+n_2+ 1)/3\rceil\), \(\lceil(n_1+n_2+n_3+ 1)/4\rceil\}\), and (ii) for every graph \(G\) with minimum degree \(\delta\) and maximum degree \(\Delta\), \(\mathrm{tvs}(G) = \max\{\lceil (\delta+\sum^i_{j=1}n_j)/(i+ 1)\rceil :i \in[\delta, \Delta]\}\), where \(n_j\) denotes the number of vertices of degree \(j\). In this paper, we disprove both of these conjectures by giving infinite families of counterexamples.